Chapter 5.8, Problem 23AYU

To determine

Find the exact value of expression.

Answer to Problem 23AYU

  $\boxed{3}$

Explanation of Solution

Given information:

Use the property of logarithm to find the exact value of expression.

  ${\log }_{2}6\cdot {\log }_{6}8$

Calculation:

Consider the given expression,

  ${\log }_{2}6\cdot {\log }_{6}8$

Apply logarithm property, change-to-base formula.

  ${\log }_{a}M=\frac{\log M}{\log a}$

Rewrite the given expression,

  $\begin{array}{l}{\log }_{2}6\cdot {\log }_{6}8={\log }_8\left(\frac{16}{2}\right)\\ {\log }_{2}6=\frac{\log 6}{\log 2},{\log }_{6}8=\frac{\log 8}{\log 6}\\ now\\ {\log }_{2}6\cdot {\log }_{6}8=\frac{\log \cancel{6}}{\log 2}\cdot \frac{\log 8}{\log \cancel{6}}\\ =\frac{\log 8}{\log 2}=\frac{\log 2^3}{\log 2}\end{array}$

Use ${\log }_a{M}^r=r{\log }_{a}M$

  $\begin{array}{l}\frac{\log 2^3}{\log 2}=\frac{3\log 2}{\log 2}\\ {\log }_{2}6\cdot {\log }_{6}8=3\end{array}$

Hence, the value of expression is $\boxed{3}$ .